Electron spin and the Stern-Gerlach experiment

A moment μ in a field B has energy U = −μ⋅B and feels a torque τ = μ × B. In a uniform field that torque makes μ precess about B at a fixed angle, and the net force −∇U is zero however strong B is. A force needs a field that changes across the atom's path: F = ∇(μ⋅B). A Stern-Gerlach magnet shapes one pole to a knife edge so that B_z grows steeply with z, giving F_z = μ_z ∂B_z/∂z. Maxwell's ∇⋅B = 0 forbids B_z from changing with z unless another component changes too, so a transverse force term appears as well. It averages away. The moment precesses about the strong B_z at γB, with γ = 1.76 × 10¹¹ rad s⁻¹ T⁻¹ for an electron spin, so in a 1 T field it turns about 2 × 10⁶ times during a 73 μs transit. The transverse projection averages to zero while μ_z stays steady, which is why the experiment measures one component and nothing else.

Atoms leave the oven with moments pointing every way. Uniform over directions means every element of solid angle, sin θ dθ dφ, is equally likely, so cos θ is uniformly distributed between −1 and +1. The probability density of μ_z = μ cos θ is then flat: P(μ_z) = 1/(2μ) for |μ_z| ≤ μ, which is normalised because ∫P dμ_z over that range is 1. Precession keeps θ fixed, so each atom holds its μ_z all the way through the magnet, and since the deflection is proportional to μ_z the screen should show one even band with no gap. The silver deposit Stern and Gerlach made in 1922 showed two separated lines with a clear gap between them. Quantum mechanics replaces the flat density with two spikes: in an unpolarised beam each atom lands in the upper or the lower spot with probability ½, and the pattern builds up one random landing at a time. No atom is ever found in the middle.

Silver has 47 electrons. Forty-six fill closed shells whose angular momenta cancel, leaving one 5s electron with l = 0 and so no orbital moment. Orbital angular momentum could not give two beams anyway: its projections mₗħ run in whole steps from −lħ to +lħ, so an even count of beams needs a half-integer angular momentum, which no integer l supplies. Stern and Gerlach read their lines through the older Bohr–Sommerfeld picture. Spin was proposed by Uhlenbeck and Goudsmit in 1925, and in 1927 Phipps and Taylor sent ground-state hydrogen, whose l = 0 is beyond dispute, through a gradient and still saw two beams. So the electron carries an extra angular momentum with 2s + 1 = 2: s = ½, S_z = ±ħ/2 and |S| = √(s(s + 1)) ħ = 0.866ħ. The largest projection is shorter than the vector, so the spin lies on a cone at 54.7° to the axis. Correspondence check: a moment of large quantum number j gives 2j + 1 spots whose spacing is 1/(2j) of the full spread, and as j grows they merge into the classical band. Spin one-half is the opposite extreme.

Inside a magnet of length L the atom feels a constant force for a time L/v, so it leaves with a sideways offset ½a(L/v)² and a sideways velocity aL/v, where a = μ_z (∂B_z/∂z)/M. A drift D to the screen adds aLD/v². Take silver, M = 1.791 × 10⁻²⁵ kg, at 550 m s⁻¹ through 800 T m⁻¹ with L = 4.0 cm and D = 6.0 cm: a = 4.15 × 10⁴ m s⁻², some 4200 times g, and each beam lands 0.439 mm off axis, 0.877 mm from the other. Run the formula backwards and a measured separation returns |μ_z|; for silver the answer is 9.28 × 10⁻²⁴ J T⁻¹, one Bohr magneton μ_B = eħ/2mₑ to within 0.12%. An orbital moment is μ_z = −μ_B mₗ, which is g = 1, so a projection of ħ/2 would carry only μ_B/2 and split the beams half as far. The spin carries twice the moment per unit of angular momentum: gₛ ≈ 2.

The integer mₗ of the previous lessons came from a boundary condition: a wavefunction must return to itself when φ goes to φ + 2π, so e^(imφ) needs integer m. Every function of position obeys that condition, so no motion of anything through space can produce mₛ = ±½. Spin is a separate, two-valued degree of freedom: the electron's state carries a two-component spinor χ alongside its ψ(r), and the spin operators act on χ alone. The spinning-ball picture fails on its own terms as well. A sphere of the classical electron radius holding ħ/2 would need its equator to move at more than a hundred times c, and collisions show no electron structure at any scale yet probed. Any rigid rotor whose charge is spread like its mass has μ = (q/2m)L exactly, which is g = 1, half the measured value. The word spin survives; the rotation does not.

Nothing in the Schrödinger equation predicts spin; it is added by hand. Dirac's 1928 relativistic equation for the electron produces it unasked: the wavefunction needs four components, carrying two spin states for the electron and two for its antiparticle, and the magnetic moment comes out with g = 2 exactly. Its low-speed limit, the Pauli equation, carries g = 2 into atomic physics. The measured value is gₛ = 2.00231930, and the extra 0.1% is the electron's coupling to the quantised electromagnetic field, through virtual photons it emits and reabsorbs. Schwinger's first-order result, aₑ = (gₛ − 2)/2 = α/2π = 0.0011614, already lands only 0.15% above the measured 0.0011597, and higher powers of α/π close the gap, one of the closest agreements between theory and measurement in physics. A Stern-Gerlach deflection cannot see the 0.1%; the Zeeman and fine-structure splittings of the next lesson carry it simply as gₛ ≈ 2.0023.